# lecture07 - MA1100 Lecture 7 Mathematical Proofs Parity of...

This preview shows pages 1–10. Sign up to view the full content.

1 MA1100 Lecture 7 Mathematical Proofs Parity of integers Divisibility of integers Direct Proofs Proof by Contrapositive Chartrand: section 3.2, 3.3, 4.1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Lecture 7 2 True Statements • Mathematical statements that are true are known as mathematical facts . • A proposition is a true mathematical statement that has a proof . • A proposition that is “ important ” is called a theorem . • A proposition that is used in the proof of a theorem is called a lemma . • A proposition that follows (easily) from a theorem is called a corollary . • There are true statements that do not have proofs e.g. definitions , axioms . Chartrand calls it “Result”
Lecture 7 3 An Example If x is an even integer, then x 2 is divisible by 4. Proposition We want to give a proof for this (universal) conditional statement. We need to : know the definitions of even integers and divisibility use some basic facts about integers use algebraic manipulation

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Lecture 7 4 Definition Examples (Mathematical) 1. Even and odd integers 2.n is divisible by m A (mathematical) definition gives the precise meaning of a word or phrase that represents some object, property or other concepts.
Lecture 7 5 Even integers 2 6 -14 Example = 2 ä 1 = 2 ä 3 = 2 ä (-7) An integer a is an even integer if there exists an integer n such that a = 2n . Definition In general, an even integer can be written in the form 2n where n is an integer

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Lecture 7 6 Odd integers 3 7 -13 Example = 2 ä 1 + 1 = 2 ä 3 + 1 = 2 ä (-7) + 1 An integer a is an odd integer if there exists an integer n such that a = 2n + 1 . Definition In general, an odd integer can be written in the form 2n + 1 where n is an integer
Lecture 7 7 Even and odd integers Why do we need definitions for even and odd integers? Is 0 an even integer? Not to identify specific even and odd integers To describe general even and odd integers To be used in the proof of statements involving even and odd integers

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Lecture 7 8 Undefined Terms Some basic concepts remain undefined . Numbers, integers, … Addition, multiplication, … Point, line, … Examples
Lecture 7 9 Divisibility Definition Let m and n be integers. We say m divides n if there exists an integer q such that n = mq .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern